What does the equal sign mean in mathematics? [duplicate]

When we say $a=a$ means that are absolutely the same right? What does equal mean in $3+2=7-2$? Does it mean that they are absolutely the same or that the value is just the same?

It is certainly true that, considered as strings of characters, the expressions "$3+2$" and "$7-2$" are different things. And it is also true that we sometimes do care about expressions from a purely formal viewpoint, and would like to distinguish between different strings of characters that mean the same thing. (For example, people who study mathematical logic and metamathematics often need to describe the properties of strings of symbols as strings.) So in such a context it would be reasonable to write something like $$\textrm{"}3 + 2 \textrm{"} \ne \textrm{"}7 - 2 \textrm{"} $$ But notice that I have here introduced an ad hoc notational convention (the quotation marks) to distinguish the expression "$3+2$" from the number that the expression stands for. In general, establishing clear notational conventions are essential if you want to distinguish between an object and an expression or string of symbols that designates that object.

This comes up at higher levels so often that we don't even pause to reflect on it. For example:

• We often distinguish between a function $f$ and the value of a function at a generic value $f(x)$, so that the equation $f(x)=g(x)$ means that two functions agree at the $x$, whereas $f=g$ means that two functions agree globally;
• We may define an equivalence relation on a structure, and use brackets or an overbar to indicate an equivalence class; then it makes perfect sense to say that $a \ne b$ but $[a] = [b]$;
• We might be interested in polynomials over a ring $R$; every polynomial $p\in R[x]$ naturally can be interpreted as a function $\hat{p}:R\to R$. However, two different polynomials may induce the same function: for example, with $R=\mathbb Z_6$ the polynomials $p=x^5 + 3x^2 + 4x$, $q = 3x^4 + 5x$ and $r = 2x$ all induce the exact same function, so $\hat{p}=\hat{q}=\hat{r}$ although $p \ne q \ne r$.

We don't normally make such fine distinctions at the level of simple arithmetic, but we could, and there may be contexts in which we should. For example if we are interested in computational complexity, it worth knowing that the expression $\sqrt{5^2 + 12^2}$ requires 4 operations, while $5+8$ requires only 1; likewise storing the first expression in memory takes more bits than storing the second expression. So while the numbers named by $\sqrt{5^2 + 12^2}$ and $5+8$ are the same number, the expressions naming them are different names.

I believe this distinction is related to what linguists call the use-mention distinction, i.e. a distinction between using a word or phrase to refer to a thing, and mentioning a word or phrase to refer to the word or phrase itself. An example from the Wikipedia page linked to in the previous sentence:

Use: Cheese is derived from milk.

Mention: "Cheese" is derived from the Old English word ċēse.

Note in this example the use of quotation marks to distinguish between the thing and the word.

Back in the days of the New Math, this kind of distinction between "number" and "numeral" (where the latter refers to the name used to express a numerical quantity) was baked into the K-12 mathematics curriculum in the United States, but that level of nuance came to be widely regarded as unnecessary hair-splitting.

When we write $$\text{LHS}=\text{RHS}$$ we mean that the expression in the left hand side is the same as that on the right.

We can split this into four categories:

• Identities: these are always true, like $$\sin^2x+\cos^2x=1,\quad x\in\mathbb{R}$$ and as you have mentioned, $$a=a$$ since for whatever value of $a$, the statement is true.

• Algebraic expressions: these include unknown parameters called variables, and the expressions are true only if the variables equate to something. For example, $$7+x=2-4\tag{1}$$ holds if, and only if, $$x=(2-4)-7=-9$$ and any other value that $x$ takes will make $(1)$ an inequality, which you can use $<,>$ or $\neq$.

• Infinite terms: It was mentioned in the comments the 'equality' $$\sin x=x+\mathcal{O}(x^3)\implies|\sin(x)-x|\le Mx^3$$ where $M$ is a constant as $x\to0$. This is clearly true since it is just a simplication of the Taylor series for $\sin$. But this is actually an inequality! Also, there is $$2=1+\frac12+\frac14+\cdots$$ where we need to write infinitely many terms of the form $1/2^n$ for the statement to hold. Note that, of course, you can express it as an infinite sum.

• Definitions: As @MarkS. has pointed out below, this is widely used to assign expressions to variables. For instance, $P=\prod_ia_i$ for a sequence $\{a_i\}$. However, there is also the symbol $:=$.